Triangulated Structures for projective Modules

TL;DR

This paper characterizes certain local rings where the category of projective modules can be structured as a triangulated category with the identity as the translation functor, satisfying all axioms including the octahedral axiom.

Contribution

It provides a characterization of local rings allowing the category of projective modules to admit a triangulated structure with the identity as translation.

Findings

Identifies conditions on local rings for triangulated structure

Establishes when the category of projective modules admits a triangulation

Includes the satisfaction of the octahedral axiom

Abstract

We give a characterisation of those local not necessary commutative rings, for which the category of projective modules admits a triangulation with the identity as translation functor. By "admits a triangulation" we mean that the category can be given the structure of a triangulated category that satisfies the standard set of axioms including the octahedral axiom.